## Table of Contents

Are you curious about the fascinating world of numbers and their properties? Today, we embark on an exciting journey to explore the number 90 and its factors. Factors play a crucial role in understanding the divisibility and unique characteristics of a number. In this comprehensive article, we will unravel the mysteries of 90 and dive deep into its factors, prime factors, pairs, common factors, and distinct factors.

Let’s begin by answering the question: “What are the factors of 90?” Factors are the numbers that divide 90 without leaving any remainder. We will provide you with a detailed list of all the factors of 90, guiding you through the process of finding them. But that’s not all! We will also explore the concept of prime factors, which are the prime numbers that divide 90. Discovering the prime factors of 90 will shed light on the unique building blocks of this intriguing number.

## What are the factors of 90?

Let’s start our exploration by answering the fundamental question: “What are the factors of 90?” We will take you on a journey through the positive integers that divide 90 evenly. You will discover the complete list of factors that make up this unique number, providing you with a solid foundation to understand its properties and relationships with other numbers.

**Let’s explore how to find the factors of 90 step by step:**

- Start with the number 1. Every number is divisible by 1, so we include it as a factor.
- Divide 90 by the next smallest number, which is 2. If the division is exact (without any remainder), then include 2 as a factor and divide 90 by 2. In this case, 90 divided by 2 equals 45 with no remainder, so we include 2 as a factor and continue to the next step.
- Repeat the process with the next smallest number, which is 3. Divide 45 by 3. Again, if the division is exact, include 3 as a factor and divide 45 by 3. In this case, 45 divided by 3 equals 15 with no remainder, so we include 3 as a factor and proceed.
- Continue this process with the next smallest numbers: 4, 5, 6, and so on, until you reach the square root of 90. In this case, the square root of 90 is approximately 9.49. Therefore, you only need to check factors up to 9.
- Divide 90 by 4. If it is an exact division, include 4 as a factor and divide 90 by 4. In this case, 90 divided by 4 is not exact (it equals 22.5), so we move on to the next number.
- Divide 90 by 5. Again, if it is an exact division, include 5 as a factor and divide 90 by 5. In this case, 90 divided by 5 is not exact (it equals 18), so we continue to the next number.
- Repeat the process with numbers 6, 7, 8, and 9. However, none of these numbers are factors of 90.
- Finally, include 10 and 90 as factors, as every number is divisible by itself.

By following these steps, you will find all the factors of 90.

Factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

## What are the prime factors of 90?

The next stop on our journey is the realm of prime factors. We will unravel the prime numbers that play a significant role in the factorization of 90. By identifying the prime factors, you will gain valuable insights into the building blocks of 90 and the fundamental elements that make it a fascinating number.

**To find the prime factors of 90, follow these steps:**

- Start with the smallest prime number, which is 2. Check if 90 is divisible by 2. If it is, divide 90 by 2 and continue dividing until you can no longer divide evenly. In this case, 90 divided by 2 equals 45 with no remainder, so we include 2 as a prime factor and move on to the next step.
- Repeat the process with the next prime number, which is 3. Divide 45 by 3. If the division is exact, divide 45 by 3 again. In this case, 45 divided by 3 equals 15 with no remainder, so we include 3 as a prime factor and proceed.
- Continue this process with the next prime numbers: 5, 7, 11, and so on, until you reach the square root of 90. In this case, the square root of 90 is approximately 9.49. Therefore, you only need to check prime factors up to 9.
- Check if 90 is divisible by 5. In this case, 90 is divisible by 5, so we divide 90 by 5. 90 divided by 5 equals 18 with no remainder. Include 5 as another prime factor.
- Check if 90 is divisible by 7. In this case, 90 is not divisible by 7.
- Repeat the process with the next prime numbers until you reach 9. In this case, 90 is not divisible by 11.

By following these steps, you can determine the prime factors of 90.

The prime factors of 90 are 2, 3, 3, and 5.

## Factors of 90 in pairs:

Prepare yourself for a thrilling exploration of factors in pairs! We will uncover the captivating combinations that result in 90. By examining the pairs of factors, you will gain a deeper understanding of how 90 can be expressed as the product of two numbers. Get ready to unlock the hidden connections between numbers and witness the beauty of factorization.

Here are the factors of 90 listed in pairs in separate tables for positive and negative factors, along with their product:

**Positive Factors:**

Factor | Pair Factor | Product |
---|---|---|

1 | 90 | 90 |

2 | 45 | 90 |

3 | 30 | 90 |

5 | 18 | 90 |

6 | 15 | 90 |

9 | 10 | 90 |

**Negative Factors:**

Factor | Pair Factor | Product |
---|---|---|

-1 | -90 | 90 |

-2 | -45 | 90 |

-3 | -30 | 90 |

-5 | -18 | 90 |

-6 | -15 | 90 |

-9 | -10 | 90 |

In the positive factors table, each factor is paired with another factor that, when multiplied together, gives a product of 90. The same goes for the negative factors table, where each negative factor is paired with another negative factor that, when multiplied together, also gives a product of 90.

## What are the common factors of 90?

As we continue our journey, we will explore the common factors of 90, shedding light on the numbers that are shared by 90 and other integers. Discover the common ground between 90 and its counterparts, and witness the power of numbers intersecting to create intriguing relationships.

Here are 10 examples of common factors of 90 with other numbers:

- Common factors of 90 and 15:

1, 3, 5, 15 - Common factors of 90 and 18:

1, 2, 3, 6, 9, 18 - Common factors of 90 and 30:

1, 2, 3, 5, 6, 10, 15, 30 - Common factors of 90 and 45:

1, 3, 5, 9, 15, 45 - Common factors of 90 and 60:

1, 2, 3, 5, 6, 10, 15, 30 - Common factors of 90 and 75:

1, 3, 5, 15 - Common factors of 90 and 9:

1, 3, 9 - Common factors of 90 and 10:

1, 2, 5, 10 - Common factors of 90 and 25:

1, 5 - Common factors of 90 and 36:

1, 2, 3, 6, 9, 18

These examples represent the numbers that can evenly divide both 90 and the given number, resulting in a remainder of 0.

## What are the distinct factors of 90?

Lastly, we will embark on an exploration of the distinct factors of 90. These factors make 90 truly distinctive, setting it apart from other numbers. By examining the unique factors of 90, you will uncover the characteristics that make it a stand-out number in the realm of mathematics.

Here are 10 examples of distinct factors of 90 with other numbers:

- Distinct factors of 90 and 15:

1, 3, 5 - Distinct factors of 90 and 18:

1, 2, 3, 9 - Distinct factors of 90 and 30:

1, 2, 3, 5, 6, 10, 15 - Distinct factors of 90 and 45:

1, 3, 5, 9, 15 - Distinct factors of 90 and 60:

1, 2, 3, 5, 6, 10, 15, 30 - Distinct factors of 90 and 75:

1, 3, 5, 15 - Distinct factors of 90 and 9:

1, 3, 9 - Distinct factors of 90 and 10:

1, 2, 5 - Distinct factors of 90 and 25:

1, 5 - Distinct factors of 90 and 36:

1, 2, 3, 6, 9, 18

These examples represent the factors that are unique to each pair, meaning they are not shared with any other factor of 90 or the given number.